Math 1b: Calculus, Series, and Di erential Equationspeople.math.harvard.edu/~jjchen/docs/spring2011-1b-worksheets.pdf · Harvard University, Spring 2011 List of Worksheets Integration

Math 1b: Calculus, Series, and Differential Equations

Harvard University, Spring 2011

List of Worksheets

IntegrationDensity and ApproximationDensity and the Definite IntegralArea and VolumeVolumes of RevolutionIntegration by SubstitutionIntegration by PartsIntegration Techniques3-dimensional Density ProblemsWorkImproper IntegralsImproper Integrals, Continued

SeriesTaylor Approximation, ContinuedTaylor SeriesDefinition of ConvergenceNth Term Test, Comparison TestGeometric Seriesp-series, Comparing Series to IntegralsAsymptoticsRatio TestPower SeriesPower Series Representations of FunctionsMore Series Problems

Differential EquationsDifferential EquationsSlope FieldsAutonomous First-Order Differential EquationsSome Methods for Solving Differential EquationsOther Ways of Looking at Differential EquationsSecond-Order Homogeneous Differential Equations with Constant Coefficientsx′′ + bx′ + cx = 0, Euler’s Formulax′′ + bx′ + cx = 0, ConcludedIntroduction to Systems of Differential EquationsPhase Plane AnalysisSystems and Shapes of TrajectoriesSystems Wrap-Up

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Density and Approximation

1. According to Wikipedia, the population density in Cambridge in the year 2000 was 15,000 people persquare mile. The area of Cambridge is about 7 square miles. What was the population of Cambridgein the year 2000?

2. A seaside village, Playa del Carmen, is in the shape of a rectangle with sides of 4 miles and 6 miles.The sea lies along a 6-mile long side. People prefer to live near the water, so the density of peopleis given by ρ(x) = 1000(20 − x2) people per square mile, where x is the distance from the seaside.(1)

Without using any calculus, how could you approximate the population of the village?

If you come up with a method, here are some questions about it:

• How accurate is your method? If you needed a quick approximation that didn’t require toomuch computation, could you come up with one? (Try!) What if you needed a really goodapproximation and were willing to spend a ton of time on computation?

• How flexible is your method? For example, if it was discovered in the next census that the densityfunction ρ(x) had changed to 500(40−√x), would your method still work?

(1)The symbol ρ is the Greek letter “rho” and is pronounced “row” (as in “Row, row, row your boat”).

1 ©Harvard Math Preceptor Group

Density and the Definite Integral

1. (Problem Set 1, #3) Pizza Pythagoras is known for its “pizza wedges,” which are shaped like righttriangles with sides of 3, 4, and 5 inches. Parmesan cheese is sprinkled on each wedge so that thedensity of cheese is given by ρ(x) ounces per square inch, where x measures distance (in inches) fromthe 3-inch side of the slice. Approximate the amount of Parmesan on each wedge.

2. People in the Boston area like to live near the city center, so the population density around Boston isρ(r) = 36,000

r2+2r+1 people per square mile, where r is the distance in miles to the center of Boston. We’dlike to find the number of people who live within 5 miles of the center of Boston.

(a) Show in a sketch how you would slice the region in question.

(b) How would you approximate the number of people who live in the k-th slice?

(c) Write a general Riemann sum that approximates the number of people who live within 5 milesof the center of Boston.

(d) By taking an appropriate limit, find a definite integral that gives the number of people wholive within 5 miles of the center of Boston.


3. A plot of land is shaped like an equilateral triangle with sides of length 2 miles. One side of the plot liesalong a river, and vegetation density varies with x, distance to the river. Suppose vegetation densityis given by ρ(x).

(a) Write a general Riemann sum that approximates the amount of vegetation in the plot.

(b) Find a definite integral that gives the exact amount of vegetation in the plot.


Area and Volume

1. Find the area under the curve y = lnx from x = 1 to x = e.

2. Find the area enclosed by y = −2x, y = x3, and y = 1.


3. (a) What shape do you get if you rotate the following region about the x-axis?

y � ��1

2Ix3+ 1M

y � 2 x - 1

-1 - ��12 ��

12

1

1

(b) Write down an integral or sum of integrals that expresses the volume of the solid from (a).


Volumes of Revolution

1. Here is one loop of the sine curve.

Π

1

(a) If you rotate this region about the x-axis, what shape do you get? What is its volume? You mayleave your answer as an integral.

(b) If you rotate the region about the horizontal line y = −1, what shape do you get? What is itsvolume? You may leave your answer as an integral.

(c) If you rotate the region about the vertical line x = 4, what shape do you get? What is its volume?You may leave your answer as an integral.


2. Let R be the region enclosed by the x-axis and the graphs of x = −1, y = −2, and y = lnx.

-2 -1 1 2 3

-2

-1

1

2

3

(a) Write an integral giving the volume generated when R is rotated about the line y = 2. (You neednot evaluate your integral.)

(b) Write an integral giving the volume generated when R is rotated about the line x = 2. (You neednot evaluate your integral.)

3. How might you describe a bagel as a solid of revolution? (That is, what sort of region would yourotate, and what line would you rotate it about?)


4. Let R be the region enclosed by the circle of radius 3 centered at the origin. Using vertical slices, findthe volume generated when R is rotated about the line x = 4. (Please evaluate your integral.)

Note: It is also possible to find the volume using horizontal slices, and you might want to try that forextra practice.


Integration by Substitution

Basic antiderivatives. You should know the following.

If n 6= −1,

∫un du = .

∫1

udu = .

∫sinu du = .

∫cosu du = .

∫eu du = .

∫1

1 + u2du = .

1. Warm-up.

(a) What isd

dte−t?

Evaluate the following integrals.

(b)

∫1

5sinu du.

(c)

∫1

etdt.

(d)

∫ 4

1

√u du.

2. Evaluate the following integrals using substitution.

(a)

∫sin(lnx)

5xdx.

(b)

∫sinx cos5 x dx.

(c)

∫ex√

2ex + 5dx.

(d)

∫ 4

1

3x2√

1 + x3 dx.


3. In this problem, you will evaluate the integral

∫ 3

0

√9− x2 dx using a clever substitution.

(a) Actually, you can easily evaluate the integral by interpreting it as a signed area. Do this.

We’ll use the substitution x = 3 sinu to do the integral. (You’ll soon see why this is a good choice.)

(b) If x = 3 sinu, what is√

9− x2 in terms of u? Simplify as much as possible. (Your final answershould not include any square root signs. If it does, you may find the trigonometric identitysin2 u + cos2 u = 1 helpful. Also, remember that

√a2 = |a|.)

The substitution x = 3 sinu is a good choice exactly because it enables us to get rid of the squareroot.

(c) If x = 3 sinu, what is dx in terms of u and du?

(d) If x = 3 sinu, what are the new endpoints of integration in terms of u?

(e) Rewrite the integral

∫ 3

0

√9− x2 dx in terms of u.

(f) Evaluate the integral in (e). (You will need the fact that

∫cos2 u du =

1

2

(u +

1

2sin 2u

)+ C,

which you will show in Problem Set 6, #3.) Does it agree with your answer to (a)?

Notice that this substitution is quite different from the ones in #2:

• In #2, we chose u so that du also appeared in the integrand. Here, we chose u just so we couldsimplify the integrand, even though du certainly didn’t appear.

• Instead of writing du in terms of x and dx, we actually did the opposite — we wrote dx in termsof u and du.

The substitution x = 3 sinu is an example of a trigonometric substitution: we substituted atrigonometric function for x even though the original integral had no trigonometric functions in it.


4. In the previous problem, we chose u for the sole purpose of getting rid of the√

9− x2, even though wedidn’t see du in the integrand. This turns out to be a handy strategy in other situations: if some part ofan integral is annoying you, call it u and see if that makes things any simpler! This is not guaranteedto help, but it’s certainly worth a try if you’re stuck.

Evaluate

∫1

2 +√x

dx.

5. Evaluate the following integrals.

(a)

∫x

x + 3dx.

(b)

∫3

9 + 16x2dx.

(c)

∫2x + 1

1 + x2dx.


Integration by Parts

Evaluate the following integrals.

1.

∫x cosx dx.

2.

∫xex dx.

3.

∫x lnx dx.

4.

∫ e

1

lnx dx.

5.

∫x2 cos 2x dx.

6.

∫ex cosx dx.


7.

∫ 1

0

arctanx dx.

8.

∫cos√x dx.

9.

∫sin 5x sin 3x dx.

10. You are given the following information about an unknown function g(x):

∫ 2

1

g(u)

udu = 3,

∫ 2

1

g(u) du = 4,

∫ 4

1

g(u) du = 5, g(1) = 2, g(2) = −2.

(a) Evaluate

∫ 2

1

(lnx)g′(x) dx.

(b) Evaluate

∫ 2

1

xg(x2) dx.


Integration Techniques

1. Which of the following is easiest to evaluate? Evaluate it.

(a)

∫5x− 4

x2 − x− 2dx.

(b)

∫5x− 4

(x− 2)(x + 1)dx.

(c)

∫5x

x2 − x− 2dx−

∫4

x2 − x− 2dx.

(d)

∫3

x + 1dx +

∫2

x− 2dx

How do the four choices relate to each other?

2. (a) Evaluate

∫1

y2 − 1dy.

(b) Evaluate

∫y2

y2 − 1dy.

3. Write down the form of the partial fraction expansion for the following integrals. (Don’t actually solvefor the coefficients.)

(a)

∫x + 5

x2 − 5x− 6dx.

(b)

∫5x2 + 7x

x2 + 4x + 3dx.


4. Evaluate

∫1

x2 + 2x + 2dx.

5. (Problem Set 6, #2) Evaluate three of the the following four integrals.

(a)

∫ln(3x + 2) dx.

(b)

∫e−x

2

dx.

(c)

∫e√x dx.

(d)

∫ln√x dx.

6. In each part, decide which method of integration you would use. If you would use substitution, whatwould u be? If you would use integration by parts, what would u and dv be? If you would use partialfractions, what would the partial fraction expansion look like? (Don’t solve for the coefficients.)

(a)

∫cosx√

1 + sinxdx.

(b)

∫(lnx)2 dx.

(c)

∫ex sinx dx.

(d)

∫x

x2 − 1dx.

(e)

∫xex

2

dx.

(f)

∫x2

x2 + 4x + 3dx.

(g)

∫et

1 + etdt.

(h)

∫arcsinx dx.


3-dimensional Density Problems

1. A cone with height 8 inches and radius 6 inches is filled with flavored slush. When the cup is heldupright with the pointed end resting on a table, the density of flavoring syrup in the cup varies withheight above the table. Suppose ρ(x) gives the number of ounces of syrup per cubic inch, where x isthe distance from the table top. Write an integral giving the total amount of syrup in the cup.

2. Suppose the density of a planet is given by the function ρ(r) =40000

1 + 0.0001r3kilograms per cubic

kilometer, where r is the distance in kilometers from the center of the planet. Find the total mass ofthe planet if its radius is 8000 km. (You need not evaluate your integral.)


3. A cylindrical candle of height 50 mm and radius 12 mm is formed by repeatedly dipping a wick ofradius 1 mm into hot wax and then allowing the new layer of wax to dry. The density of each newlayer is slightly different, so the density of the candle varies with the distance to the wick. Supposeρ(x) gives the density in grams per cubic mm of the wax, where x measures the distance to the wick(in mm).

(a) Write an integral giving the mass of the candle. (The mass of the wick is negligible.)

(b) Show that, if ρ(x) = 1x+1 , then the mass of the candle is 1100π grams.

4. We can model a muffin as a solid of revolution, obtained by rotating the following region about they-axis. Due to a poor recipe, the chocolate chips in our muffin tend to sink to the bottom. The amountof chocolate in the muffin is given by ρ(y) = 5−y grams per cubic inch, where y represents the distanceto the bottom of the muffin. Find the total amount of chocolate in the muffin.

y � 3 - ��x2

4

y � 2 x - 2

1 2 3 4

1

2

3

4


Work

Definition. If a constant force F acting in the direction of motion of an object causes a displacement d,then the work done by the force on the object is defined to be the product of the force and displacement;that is, work = force · displacement.

1. For no particular reason, you decide to lift a very heavy rock. If the rock has a mass of 40 kg and youlift it 2 m, how much work do you do in lifting it?

2. A 50 m chain with mass density 2 kg/m is attached to a drum hung from the ceiling. The ceilingis high enough so that the bottom of the chain doesn’t touch the floor. Write an integral giving theamount of work required to wind the chain around the drum. (Assume that the height of the drum isnegligible.) You may express your answer in terms of the gravitational constant g, and you need notevaluate your integral.


3. A 100 kg bag of sand is hoisted 50 m at a rate of 5 m/s. Because of a hole in the bag, 2 kg of sand islost each second. Find the work done.

4. An ant-hill in the Amazon rainforest has the shape of a perfect hemisphere, with its base on the forestfloor. The ant-hill has a 1 m radius and a uniform density of 50 kg/m3. How much work have the antsdone to assemble this ant-hill, lifting materials from the level of the forest floor? You may leave youranswer as an integral in terms of g.


Improper Integrals

1. (a) Does

∫ ∞

1

1

x2dx converge or diverge? If it converges, evaluate it.

(b) Does

∫ ∞

1

1

xdx converge or diverge? If it converges, evaluate it.

2. Using #1, can you conclude anything about whether the following integrals converge or diverge? (Tryto figure this out without evaluating the integrals!)

(a)

∫ ∞

1

1

x3dx? (b)

∫ ∞

1

1

x1/2dx? (c)

∫ ∞

1

1

x3/2dx?


3. In each of the following, decide whether the improper integral converges or diverges, and use theComparison Theorem to justify your answer.

(a)

∫ ∞

1

x√3 + x5

dx. (b)

∫ ∞

1

1

x2 +√x

dx.

4. Does the improper integral

∫ ∞

0

sinx dx converge or diverge? If it converges, what is its value? How

about

∫ 0

−∞sinx dx?

5. Decide whether the following improper integrals converge or diverge. Explain your reasoning thor-oughly.

(a)

∫ ∞

5

1

xdx.

(b)

∫ ∞

4

x2

x3 − 1dx.

(c)

∫ ∞

1

e−1/x

x3dx.


Improper Integrals, Continued


∫ ∞

−∞

x

1 + x2dx converge or diverge? If it converges, to what number does

it converge?

-20 -10 10 20

y � ��x

x2+ 1

2. True or false: For any function f(x),

∫ ∞

−∞f(x) dx = lim

a→∞

(∫ a

−af(x) dx

).


∫ ∞

−1

1

x2dx converge or diverge? If it converges, to what number does it

converge?


An integral is improper if

1. the interval of integration is unbounded and/or2. the integrand is unbounded somewhere on the interval of integration

The basic strategy for dealing with an improper integral is:

1. Identify all improprieties.2. If necessary, split up the integral into a sum of integrals so that every impropriety is an endpoint

of integration and each new integral has at most one impropriety.3. Compute each of these new improper integrals as a (one-sided) limit of definite integrals.4. If any one of the new integrals diverges, the original improper integral diverges. If all of the new

integrals converge, sum to find out what the original converges to.

Z Caution: This means that divergent summands can never “cancel” one another!

4. Carry out steps 1 and 2 of the above strategy on the following integrals.

(a)

∫ 2

−2

x

x2 − 1dx.

(b)

∫ ∞

−∞sinx dx.

5. Suppose f(x) is a function which is defined, continuous, and positive on [1,∞), and you are told that

limx→∞

f(x) = 1. Does

∫ ∞

1

f(x) dx converge or diverge, or is there not enough information to tell?


Determine whether the following integrals converge or diverge. Explain your reasoning. (You may use theabove strategy to actually compute the integral or use the Comparison Theorem to compare to an integralthat is easier to compute.)

6.

∫ ∞

0

e−x2

dx.

7.

∫ ∞

1

1

x4 + 2dx.

8.

∫ ∞

0

1

ex + xdx.

9.

∫ ∞

−∞

1

x3dx.


Taylor Approximation, Continued

1. If you want to find the degree n Taylor polynomial approximation a0 + a1(x− c) + a2(x− c)2 + · · ·+an(x− c)n for f(x) centered at c, write a formula for the coefficient ak.

2. Find the degree 15 Taylor polynomial approximation for f(x) = cosx centered at 0.

3. Rewrite the following sums in · · · notation. (This means: Write down at least the first 3 terms + · · ·+the last term.)

(a)

100∑

k=1

1

2k.

(b)

15∑

k=3

(−1)k

k.

(c)

25∑

k=0

x2k+1

k + 1.

(d)

26∑

k=1

x2k−1

k.


4. Rewrite the following sums in sigma notation.

(a)1

33+

1

43+

1

53+

1

63+ · · ·+ 1

1003.

(b)√

2−√

4 +√

6−√

8 +√

10− · · · −√

120.

(c) 1 + x3 + x6 + x9 + x12 + x15 + · · ·+ x30.

(d) x− x3 + x5 − x7 + x9 − · · ·+ x101.

(e) x− x3

2+

x5

3− x7

4+ · · · − x31

16.

5. (a) Rewrite your answer to #2 in sigma notation.

(b) Find the degree 25 Taylor polynomial approximation for f(x) = cosx centered at 0. Write it inboth sigma and · · · notation.

(c) Use the degree 25 Taylor polynomial approximation to get an approximation for cos 0.1.


Taylor Series

1. What is the Taylor series expansion of cosx centered at 0? Write it in both sigma and · · · notation.

2. What is the Taylor series expansion of sinx centered at 0? Write it in both sigma and · · · notation.


We hope that, by using a “polynomial of infinite degree,” we end up with something that is not just anapproximation for our function but is actually equal to the function. We don’t really know if this is true yet,but let’s see how it would be useful to have such an alternate representation of a function.

For the remaining problems, take on faith that sinx is actually equal to its Taylor series expansion about 0.

3. Write an “infinite polynomial” representation of:

(a) sinx3.

(b)

∫sinx3 dx. (Note: The antiderivatives of sinx3 are not elementary functions!)

4. Find the degree 100 Taylor polynomial approximation of sinx3.

5. Let f(x) = sinx3. What is f (100)(0)? f (51)(0)?


Definition of Convergence

1. Does the series 1 + 1 + 1 + · · · converge or diverge? If it converges, what is its sum?

2. Does the series 1 + (−1) + 1 + (−1) + · · · converge or diverge? If it converges, what is its sum?

3. (a) Does the series 1 +1

2+

1

4+

1

8+ · · · converge or diverge? If it converges, what is its sum?

(b) Draw a diagram showing the first 6 partial sums of the series 1 +1

2+

1

4+

1

8+ · · · .


4. In this problem, you’ll investigate the series∞∑

k=1

2

(2k)2 − 1.

(a) Write out the first 4 terms of the series.

(b) Find the first 4 partial sums of the series. Express them as simplified fractions.

(c) Looking at your answer to (b), guess a formula for the n-th partial sum.

(d) Based on your guess from part (c), does the series converge or diverge? If it converges, what isits sum?

5. (a) Draw a partial sum diagram showing the first 6 partial sums of the series

∞∑

k=0

(−1

2

)k

.

(b) Based on your diagram, do you think the series∞∑

k=0

(−1

2

)k

converges or diverges?

6. Here is a diagram showing the first 6 partial sums of an unknown series∞∑

k=1

ak.

s1 s2 s3s4 s5 s6

- ��

12 - ��

14

0 ��

14 ��

12

(a) From the diagram, determine whether each of a1, a2, . . . , a6 is positive, negative, or zero.

(b) Which of a1, a2, . . . , a6 is biggest in magnitude?


Nth Term Test, Comparison Test

1. Suppose you know that the series∞∑

k=1

ak converges. Let sn = a1 + a2 + · · · + an, the n-th partial sum.

For each of the following, determine whether the statement must be true, must be false, or could beeither true or false.

(a) limn→∞

an = 0.

(b) limn→∞

sn = 0.

(c)∞∑

k=5

ak converges.

2. Suppose a1, a2, . . . are numbers and limk→∞

ak = 0. Does it necessarily follow that the infinite series

∞∑

k=1

ak converges?

3. The series∞∑

k=1

1

k= 1 +

1

2+

1

3+

1

4+ · · · is known as the harmonic series.

(a) What does the Nth Term Test tell you about the harmonic series?

(b) Does the harmonic series converge or diverge?


4. (a) Does the series 1 +1

2+

1

2+

1

4+ · · · +

1

4︸︷︷︸4 terms

+1

8+ · · · +

1


+1

16+ · · · +

1

16︸︷︷︸16 terms

+ · · · converge or diverge?

(b) Does the series1

2+

1

4+

1

4+

1

8+ · · · +

1


+1

16+ · · · +

1

16︸︷︷︸8 terms

+1

32+ · · · +

1

32︸︷︷︸16 terms

+ · · · converge or diverge?

5. Does the series∞∑

k=1

1

2k + kconverge or diverge?

6. Does the series∞∑

k=1010

1

100000000kconverge or diverge?

7. Decide whether each of the following series converges, and use the Comparison Test to justify youranswer mathematically.

(a)∞∑

k=100

5 + 3 sin k

k. (b)

∞∑

k=37

5 + 3 sin k

2k.


Geometric Series

1. If you suffer from allergies, your doctor may suggest that you take Claritin once a day. Each Claritintablet contains 10 mg of loratadine (the active ingredient). Because of the way the human bodymetabolizes loratadine, no matter how much is in your body at a given time, only 1

8 of that amount

remains 24 hours later.(1)

(a) If you take one Claritin tablet each morning for 2 successive days, how much loratadine is in yourbody right after you take the 2nd tablet?

(b) If you take one Claritin tablet every morning for a week, how much loratadine is in your bodyright after you take the 3rd tablet? 7th tablet? (Don’t try to simplify your computations; justwrite out an arithmetic expression.)

(c) If you take Claritin for years and years, will the amount of loratadine in your body level off? Orwill your bloodstream be pure loratadine?

(1)This estimate comes from the fact that the average half-life of loratadine is known to be 8 hours.


Definition. A geometric series is a series that can be written in the form∞∑

k=0

ark = a+ar+ar2+ar3+· · · .

2. Let sn be the n-th partial sum of the geometric series a + ar + ar2 + ar3 + · · · . That is, sn is the sumof the first n terms. In this problem, we’ll consider only geometric series with a 6= 0.

(a) Write a closed form expression for sn if r = 1. (A “closed form expression” is one without · · · or∑; if you had values for a and n, you could type it directly into a calculator to evaluate it.)

(b) Write a closed form expression for sn, assuming r 6= 1.

(c) For what values of a and r does the geometric series a+ar+ar2 + · · · converge? For what valuesdoes it diverge? When it converges, what is its sum?

This result is incredibly important; you should feel free to cite it from now on. (When using it, besure to explain what a and r are for your geometric series.)


3. (a) For what values of x does the geometric series∞∑

k=0

xk = 1 + x + x2 + · · · converge? For what

values of x does it diverge? When it converges, what does it converge to?

(b) How would you graph∞∑

k=0

xk?

4. Remember that a geometric series is a series of the form a + ar + ar2 + · · · =∞∑

k=0

ark. Which of the

following series are geometric? For each geometric series, identify a and r, and say whether the seriesconverges or diverges; if it converges, find its sum.

(a)∞∑

k=1

(−1)k2k+1

3k.

(b)∞∑

k=1

1

k3.

(c)∞∑

n=4

2

32n.


5. The following figure is called the Koch snowflake.

Here’s how it is made. Let P0 be an equilateral triangle whose sides have length 1. Replace each linesegment in P0 with this shape (called the “motif”). Each line segment in the motif is 1/3 of the lengthof the original line segment.

This gives you a new polygon P1; each line segment in P1 has length 1/3. To get P2, replace each linesegment in P1 with the motif. Now, each line segment in P2 has length 1/9. If you do this again, youget P3, which is composed of line segments of length 1/27. If you continue this process forever, youend up with the Koch snowflake.

P0 P1 P2 P3 P4

(a) How many sides does polygon Pk have?

(b) What is the length of each side in polygon Pk?

(c) Write a series which expresses the total area of the Koch snowflake.

(d) What is the area of the Koch snowflake?

(e) What is the perimeter of polygon Pk? What does this tell you about the perimeter of the Kochsnowflake? (This question has nothing to do with series.)


p-series, Comparing Series to Integrals

1. Does∞∑

k=1

1

k2converge or diverge?

2. We have already seen that the harmonic series

∞∑

k=1

1

kdiverges. (We showed this by comparing to

another series.) Can you show this by comparing to an improper integral?

3. For what values of p is the p-series

∞∑

k=1

1

kpconvergent? For what values of p is it divergent?

From now on, you may simply cite the result of this problem. When you do so, be sure to identify thevalue of p you are considering.


4. Now that you know about geometric series and p-series, you can use the Direct Comparison Test inmany more situations than you could before.

Use the Direct Comparison Test to explain why the following series converge or diverge.

(a)∞∑

k=1

| cos k|k3

.

(b)∞∑

k=1

3 + sin k

0.9k.

5. Here is a new definition: a series∑

ak is called absolutely convergent if the series∑ |ak| is convergent.

Decide whether or not each of the following series is absolutely convergent. (Notice that, when youdecide whether

∑ak is absolutely convergent, you’re not looking at

∑ak at all; instead, you’re looking

at the series∑ |ak|, whose terms are all non-negative.)

(a)∞∑

k=1

cos k

k3.

(b)∞∑

k=1

(−1)k+1 3 + sin k

0.9k.


Asymptotics

1. Does the series

∞∑

k=1

3

2k − 100converge or diverge?

Definition. We say that bk grows faster than ak as k →∞ if limk→∞

akbk

= 0 or, equivalently, limk→∞

bkak

=∞.

We write this as ak � bk.

Here are some common expressions listed according to their growth rates:

ln k � km � kn � ak � bk � k!� kk where 0 < m < n and 1 < a < b (1)

Here are some more useful facts:

• A sum is asymptotic to its “largest” term (where largest means the one that grows fastest).

• If bk � ak, then bk � cak for any constant c. For instance, since k3 � k2, k3 � 50000k2.

2. Asymptotically simplify the following expressions as much as you can. (That is, find a simpler ex-pression that is asymptotic to the given one, and try to make your simpler expression as simple aspossible.)

(a)k1000 + k999 + 1.1k

ln k + k4.

(b)k! + πk17 + 10000k

1010 ln k + kk + k!.

(c)2k(k3 + k2)

3k.

(d)1

k!(k9 − 7k2)5/3.


3. Use asymptotics to help you decide whether the following series converge or diverge.

(a)

∞∑

k=1

5√k

k2 + 2k.

(b)∞∑

k=1

k4 + 2k − 7 ln k

(3k − 10k2)k!.

4. Asymptotically simplify each expression.

(a)5 + 3 sin2 k

k3 + k!.

(b)(k + sin k) ln k

5k · k3 .

5. Decide whether each of the following series converges, and justify your conclusion mathematically.

(a)∞∑

k=1

k3(e− 2 cos k)√k7 − 5k

(b)

∞∑

k=1

k2

(100 + 5k2 + k11)1/3


Ratio Test

Definition. A series∑

ak is called absolutely convergent if the series of absolute values∑ |ak| is

convergent.

Theorem. If a series∑

ak is absolutely convergent, then it is convergent.

The opposite (converse) of the theorem is not true: there are convergent series which are not absolutelyconvergent. These series are called conditionally convergent.

1. (a) Suppose you have a series

∞∑

k=1

ak whose terms are nonzero (but the series could have both positive

and negative terms), and you know that limk→∞

∣∣∣∣ak+1

ak

∣∣∣∣ = 5. What does this tell you about the

magnitude (absolute value) of the terms as k gets really big? Does the series converge or diverge,or is there not enough information to tell?

(b) Suppose you have a series∞∑

k=1

bk whose terms are all positive, and you know that limk→∞

bk+1

bk=

2

3.

Does the series converge or diverge, or is there not enough information to tell?

(c) Suppose you have a series

∞∑

k=1

ck whose terms are nonzero (but the series could have both positive

and negative terms), and you know that limk→∞

∣∣∣∣ck+1

ck

∣∣∣∣ =2

3. Does the series

∞∑

k=1

|ck| converge or

diverge, or is there not enough information to tell? What does this tell you about the series∞∑

k=1

ck?


2. What does the Ratio Test tell you about the following series?

(a)∞∑

k=1

k

3k.

(b)∞∑

k=0

(−1)k+1 1000k

k!.

(c)

∞∑

k=1

1

k.

(d)∞∑

k=1

(−1)k+1 1

k.

(e)∞∑

k=1

1

k2.


3. (a) What is the Taylor series expansion of ex about 0?

(b) What does the Ratio Test say about the convergence of the series you found in (a)?

4. What does the Ratio Test tell you about the series∞∑

k=1

(3k)!

52k?


Power Series

1. In Problem Set 20, #5, you looked at the Taylor series of ln(1 + x) centered at 0,∞∑

k=1

(−1)k+1xk

k=

x− x2

2+

x3

3− x4

4+ · · · .

(a) What did the Ratio Test tell you about this Taylor series?

(b) For what values of x was the Ratio Test inconclusive? How did you determine whether this Taylorseries converged for those values of x?

(c) What are all values of x for which this Taylor series converges?

2. A long time ago (Problem Set 14, #2), you found that the Taylor series of1

1− xcentered at 0 is

∞∑

k=0

xk = 1 + x + x2 + · · · .

(a) The series∞∑

k=0

xk happens to be geometric. For what values of x does it converge? For what

values of x does it diverge? When it converges, what does it converge to?

(b) Graph∞∑

k=0

xk.


A power series centered at the number c is a series of the form∞∑

k=0

ak(x− c)k = a0 + a1(x− c) + a2(x−

c)2 + · · · where x is a variable and the ak are constants.

3. Which of the following are power series?

(a)∞∑

k=1

√k(x− 3)1+2k.

(b)

∞∑

k=1

√k(x− 3)1−2k.

(c)∞∑

k=0

√x(x− 5)k.

(d)∞∑

k=5

(x− k)k

k.

Theorem on convergence of a power series. For a given power series∞∑

k=0

ak(x− c)k centered at c,

there are 3 possibilities:

1. The series converges only when x = c.

2. The series converges absolutely for all x.

3. There is a positive number R such that the series converges absolutely when |x − c| < R anddiverges when |x − c| > R. R is called the radius of convergence. (Note that this doesn’t sayanything about what happens when |x− c| = R.)

If (1) is true, we say that the radius of convergence of the series is 0; if (2) is true, we say that the radiusof convergence is ∞. So, every power series has a radius of convergence.

Terminology. The interval of convergence of a power series is the set of x for which the power seriesconverges. The largest open interval of convergence is the open interval (c−R, c + R).

4. (a) What is the radius of convergence of the power series∞∑

k=1

(−1)k+1xk

k(the power series from #1)?

What is the interval of convergence? What is the largest open interval of convergence?

(b) What is the radius of convergence of the power series

∞∑

k=0

xk (the power series from #2)? What

is the interval of convergence? What is the largest open interval of convergence?


5. (a) Find the radius of convergence of the power series∞∑

k=1

(x− 3)k

k2 · 7k .

(b) Find the largest open interval of convergence of the power series∞∑

k=1

(x− 3)k

k2 · 7k .

(c) Find the interval of convergence of the power series∞∑

k=1

(x− 3)k

k2 · 7k .

(d) Draw the interval of convergence on a number line.

Definition. If a function f(x) is equal to a power series∞∑

k=0

ak(x − c)k (possibly only on a limited

domain around c), we say that∞∑

k=0

ak(x− c)k is a power series representation of f(x) (at c).

6. (a) Find a power series representation of the function f(x) =1

1 + 4x2. For what x can we be sure

that this representation is valid?


(b) Find a power series representation of the function g(x) =x

1 + 4x2. For what x can we be sure

that this representation is valid?

(c) In fact, the series you found in the two previous parts should have both been geometric. Use yourknowledge of geometric series to verify that your answers are correct.

7. Find a power series representation ofx3

1 + 27x3. For what x can we be sure that this representation is

valid?

8. In this problem, you’ll look at the power series∞∑

n=0

n!xn.

(a) Write out the first few terms of the power series. Does the series converge when x = 0?

(b) Find all values of x for which the series∞∑

n=0

n!xn converges. (Make sure your answer is consistent

with your answer to #8.)

(c) What is the radius of convergence of this power series?


Power Series Representations of Functions

Theorem. If the power series

∞∑

k=0

ak(x−c)k = a0 +a1(x−c)+a2(x−c)2 + · · · has radius of convergence

R where R > 0 or R =∞, then the function f(x) =∞∑

k=0

ak(x− c)k = a0 + a1(x− c) + a2(x− c)2 + · · ·

is differentiable on (c−R, c + R). On that interval,

1. f ′(x) =∞∑

k=1

kak(x− c)k−1 = a1 + 2a2(x− c) + 3a3(x− c)2 + · · · .

2.

∫f(x) dx = C +

∞∑

k=0

akk + 1

(x− c)k+1 = C + a0(x− c) +a12

(x− c)2 +a23

(x− c)3 + · · · .

The power series in (1) and (2) both have radius of convergence R. (Note: Although the radius ofconvergence remains unchanged, the interval of convergence may change.)

1. Find a power series representation of1

(1− x)2centered at 0. (Hint:

1

(1− x)2is the derivative of

1

1− x.) What is the radius of convergence of the power series you have found?

2. (a) Find a power series representation for ln(1 + x) centered at 0. (Hint: What is the derivative ofln(1 + x)?)

(b) What is the radius of convergence for the power series you have found? What is the interval ofconvergence?


(c) You showed in Problem Set 15, #5 that the alternating harmonic series 1 − 1

2+

1

3− 1

4+ · · ·

converges. What is its sum?

Theorem. If f(x) has a power series representation centered at c (that is, f(x) =∞∑

k=0

ak(x − c)k for

|x− c| < R), then that power series must be the Taylor series of f at c.

3. (a) Taking for granted that sinx =∞∑

k=0

(−1)kx2k+1

(2k + 1)!= x− x3

3!+

x5

5!− · · · for all x, find the Taylor

series of f(x) = x sin(x3) at 0.

(b) What is the radius of convergence of the power series you found in (a)?

(c) What is the degree 15 Taylor polynomial approximation of f(x) centered at 0? (Please write outall terms.)

(d) What is f ′′′(0)? f (4)(0)?


4. In Problem Set 14, #1, you found that the Taylor series of ex centered at 0 is∞∑

k=0

xk

k!= 1+x+

x2

2!+ · · · .

(a) What is the interval of convergence of the power series?

(b) Define f(x) =∞∑

k=0

xk

k!= 1 +x+

x2

2!+ · · · on the interval you found in (a). (That is, the domain of

f is the interval of convergence of the power series.) Write a power series representation of f ′(x).Please give your answer in both sigma and · · · notation. For what x is this representation valid?

(c) It is a fact(1) that there is exactly one function h(x) such that h′(x) = h(x) and h(0) = 1. Usingthis fact, show that ex is equal to its Taylor series.

From now on, you may use the fact that ex is equal to its Taylor series for all x.

(1)We will understand this fact better when we study differential equations.


5. Consider the function f(x) =1

1− 4x2.

(a) Find a power series representation of

∫1

1− 4x2dx. For what x is your representation valid?

(Give the largest open interval.)

(b) If possible, use your answer to (a) to write down a series that converges to

∫ 1.5

0

1

1− 4x2dx. If it

is not possible, explain why.


More Series Problems

1. (a) Give a power series representation of ex. For what x is it valid? (How do we know that the powerseries is actually equal to ex?)

(b) Use (a) to find a series whose sum is e.

(c) Explain why the series in (b) converges.

(d) Suppose you use the first 5 terms of the series you found in (b) to approximate e. Write a seriesthat gives the exact error in this approximation.

(e) By comparing the error to a geometric series, put an upper bound on the error (that is, find anumber that you can demonstrate is bigger than the error). Try to make your upper bound asclose as possible. (After all, it’s a lot more useful to say the error is at most 0.1 than it is to saythat the error is at most 1000).

Hint: First factor out 15! .


(f) Suppose you instead use the first 7 terms of the series to approximate e. Find an upper boundon the error.

(g) Suppose you use the first n terms of the series to approximate e. Find an upper bound on theerror.

(h) Suppose you want to approximate e with an error of at most 10−5. Explain how you could usethe previous part to find such an approximation.

2. (a) Find the degree 2 Taylor polynomial approximation of (1 + x)1/3 centered at 0.

(b) Does it seem like a good idea to use the approximation you’ve just found to approximate 1.011/3?10011/3? Why or why not?


(c) Let q be any constant. Find the degree 2 Taylor polynomial approximation of (1 + x)q centeredat 0.

(d) What happens when q = −1? Is this what you expect?

(e) What happens when q = 1? q = 2?

3. Let f(x) =∞∑

k=0

(x + 4)2k+1

(k + 3)9k.

(a) Find the radius of convergence of the power series defining f .


(b) Which of the following quantities are defined? Which are easy to compute exactly? Which canyou easily find a series representation of?

i. f(6).

ii. f(−6).

iii. f (50)(−2).

iv. f (100)(0).

v. f (25)(−4).

vi. f (10)(4).

vii. f ′(−3).

viii.

∫ 0

−2f(x) dx.

ix.

∫ −3

−4f(x) dx.


Differential Equations

1. Mr. Moneybags decides to open a bank account with an opening deposit of $1000. Suppose thatthe account earns a nominal annual interest rate of 6%, compounded annually.(1) Assuming Mr.Moneybags completely ignores the account after he opens it (so he doesn’t make any deposits orwithdrawals), how much money does the account have t years after Mr. Moneybags opens it? Can yougraph his money vs. time?

2. Suppose that Mr. Moneybags had instead deposited his money in a bank that offered monthly com-pounding (but everything else was the same). That is, the bank has 12 compounding periods a year.How much money would the account have after t years?

3. Continuous compounding is defined to be the limit as n→∞ of having n compounding periods a year.If Mr. Moneybags had used a bank with continuous compounding, how much money would he haveafter t years?

(1)For economists, there are two uses of the phrase “nominal interest rate”; one is trying to account for the effects of inflation.We’re using it in the other sense, which will be easier to understand a bit later.


4. In the situation of the previous problem, we say that Mr. Moneybags gets a “6% nominal annualinterest rate, compounded continuously.” If we are modeling his money M(t) as a continuous function,what is the relationship between dM

dt and M?

5. $1000 is deposited in a bank account. The account has a nominal annual interest rate of 6%, com-pounded continuously. Money is being deposited continuously at a rate of $600 per year.(2) How couldyou model this situation?

In #6 - #9, write a differential equation that reflects the situation. Include initial conditions if the informa-tion is given.

6. The population of a certain country increases at a rate proportional to the population size. LetP = P (t) be the population at time t.

7. A yellow rubber duck is dropped out of the window of an apartment building from a height of 80m. Let s = s(t) be the height of the duck above the ground at time t. (Gravity is the acceleration

−9.8 m/s2.)

Hint: Write an equation involving a second derivative.

(2)In reality, you cannot deposit money continuously, but it’s convenient to use a continuous model.


8. Ferdinand is trying to fill a bucket from a faucet. Unfortunately, he doesn’t realize that there is a smallhole in the bottom of the bucket. Water flows in to the bucket from the faucet at a constant rate of0.75 quarts per minute, and it flows out of the hole at a rate proportional to the amount of water W (t)already in the bucket (due to the increased water pressure).

9. (a) Harry Potter is practicing his magical Refilling Charm on a mug of pumpkin juice. He uses theRefilling Charm to add juice to the mug at a rate inversely proportional to the amount of juicecurrently in the mug. Let J(t) be the amount of juice in the mug at time t.

(b) Harry decides to put his Refilling Charm into action while Ron is drinking a mug of pumpkinjuice. Ron drinks at a constant rate of 50 ml/min, while Harry uses his Refilling Charm as in theprevious part. Let J(t) be the amount of pumpkin juice in Ron’s mug at time t.

The following problems are about solutions to differential equations.

10. Which of the following is a solution to dydx = y?

i. y = x2

2 + C.

ii. y = ex + C.

iii. y = Cex.

11. Give two solutions to dydx = 5y. What is the general solution?


12. Give two solutions to to dydx = 5x. What is the general solution?

13. Which of the following is the solution ofdM

dt= 0.06M + 600 satisfying M(0) = 1000?

i. M(t) = 1000e0.06t + 600t

ii. M(t) = 11,000e0.06t − 10,000

iii. M(t) = (1000 + 600t)e0.06t

14. Show that, if r is any constant, then limn→∞

(1 +

r

n

)n

= er.


Slope Fields

1. (Problem Set 24, #8) Martha Stewart is mixing up a big batch of eggnog for the holidays. Supposeshe has a big mixing vat containing 100 gallons of egg yolks. At time t = 0, her assistant starts pouringa milk and sugar mixture containing 1/2 lb of sugar per gallon into the vat at a rate of 4 gallons perminute. At the same time, mixed eggnog starts leaving the vat a rate of 4 gallons per minute.

Write a differential equation for the amount S(t) of sugar (measured in pounds) in the vat at time t.Can you write down an initial condition as well?

2. Suppose instead that, in the previous problem, mixed eggnog left the vat at a rate of 6 gallons perminute. How would this change the differential equation and initial condition?

3. Draw slope fields for the following differential equations:

(a)dy

dt= 1. (b)

dy

dt= −t.


(c)dy

dt= −y. (d)

dy

dt= − t

y.

Now, go back and try to figure out the general solution of each differential equation.

4. (a) If you have a differential equation of the form dydt = f(t) (that is, dy

dt depends only on t, not on y;

an example is dydt = t2 + 1), what can you say about its slope field? How do different solutions of

the differential equation relate to each other?

(b) If you have a differential equation of the form dydt = f(y) (that is, dy

dt depends only on y, not ont), what can you say about its slope field? How do different solutions of the differential equationrelate to each other?

5. (a) Draw the slope field for the differential equation dydt = 1−y. Sketch two solutions to the equation.

(b) Which of the following is a solution to dydt = 1 − y?

i. y = Ce−t

ii. y = Cet − t

iii. y = Ce−t − 1

iv. y = Ce−t + 1

v. y = Cet + 1


6. Here are four slope fields.

0 10 20 30 40 50t0

25

50

75

100S

0 10 20 30 40 50t0

25

50

75

100S

(A) (B)

0 10 20 30 40 50t0

25

50

75

100S

0 10 20 30 40 50t0

25

50

75

100S

(C) (D)

(a) Which is the slope field of the differential equation you found in #1? On that slope field, sketchthe solution corresponding to your initial condition from #1. Over time, what happens to theamount of sugar in the vat?

(b) Which is the slope field of the differential equation in #2? On that slope field, sketch the solutioncorresponding to your initial condition. Over time, what happens to the amount of sugar in thevat?


7. (a) Martha Stewart is still busy perfecting her 100 gallons of eggnog. Unfortunately, one of herassistants messed up and dumped too much sugar in the vat (Martha herself would never make amistake), so the eggnog in the tank has a concentration of 3

4 pounds of sugar per gallon. To fixthe problem, Martha Stewart starts adding unsweetened eggnog to the vat at a rate of 1 gal/min.Unfortunately, another clumsy assistant punches a hole in the vat, so the mixed eggnog is drippingout of the vat at the rate of 3 gal/min.

Let t denote the time in minutes, with t = 0 the time that Martha Stewart starts pouringunsweetened eggnog into the vat. Let S(t) be the amount of sugar in the vat at time t, measuredin pounds. Write a differential equation and initial condition for S(t).

(b) Which of the four slope fields in #6 is the slope field of the differential equation you found? Onthat slope field, sketch the solution corresponding to your initial condition. Over time, whathappens to the amount of sugar in the vat?


Autonomous First-Order Differential Equations

1. Here are three slope fields.

-2 -1 1 2t

-2

-1

1

2

y

-2 -1 1 2t

-2

-1

1

2

y

-2 -1 1 2t

-2

-1

1

2

y

(I) (II) (III)

For each of the following differential equations, find the picture above which shows its slope field.

(a)dy

dt= t5 − 4t3 + 4t− 1

2. (b)

dy

dt= y5 − 4y3 + 4y − 1

2.

2. Use qualitative analysis to sketch a family of representative solutions for each differential equation.Identify the equilibrium solutions of each differential equation, and say whether they are stable orunstable.

(a)dy

dt= (y + 1)(4 − y). (b)

dy

dt= (y + 1)(y − 2)2.


Qualitative Analysis ofdy

dt= f(y) or How can we sketch solutions while being as lazy as possible?

So far, when we’ve wanted to sketch solutions of a differential equation, we’ve sketched its slope field.Qualitative analysis gives us the same information when our differential equation has the form dy

dt = f(y),with a lot less work! The basic method is:

1. Find the equilibrium solutions and sketch these.

2. The equilibrium solutions divide the picture into horizontal strips. In each strip, test one pointto determine whether dy

dt is positive or negative. This tells you whether solutions in that stripincrease or decrease, and then you can draw a representative solution in that strip.

If you want to draw the solution satisfying an initial condition, then it is just a horizontal shift of oneof your representative solutions.

3. For each picture, think of an autonomous first-order differential equation whose solution curves looklike those in the picture.

(a)

t

1

2

3

4

y


(b)

t

1

2

3

4

y

(c)

t

1

2

3

4

y


(d)

t

10

20

30

40y

(e)

t

-10

-5

5

10

15

y


(f)

t

-10

-5

5

10

15

y

(g)

t

-2

-1

1

2

3

y


(h)

t

-3

-2

-1

1

2

y


Some Methods for Solving Differential Equations

1. Find the general solution of the differential equationdM

dt= 2.4 − 0.2M .

2. Solve the differential equationdy

dt= − t

yusing separation of variables. (Previously, we solved by draw-

ing the slope field, guessing the solution, and checking that it worked by plugging into the differentialequation.)

3. Decide whether each of the following differential equations is separable.

(a)dy

dt= t + y.

(b)dy

dt=

y

sin t.

(c)dy

dt=

sin t

y+ t.


4. In this problem, we’ll solve the non-separable differential equation y′+y = te−t by using a substitutionu = ety.

(a) You are given a substitution u(t) = ety(t); rewrite this equation to solve for y(t).

(b) Use your answer to (a) to rewrite the differential equation y′ + y = te−t in terms of u (and t).

(c) Solve the differential equation you found in (b). (Find all solutions.)

(d) Use your answer to the previous part to solve for y(t).

(e) Find the solution of y′ + y = te−t satisfying the initial condition y(0) = −1.

5. Solve the differential equation y′ + 2xy2 = 0. (Find all solutions.)

6. Solve the differential equation y′ =y2 + xy

x2by using the substitution u =

y

x.

7. Solve the differential equationdy

dt=

t2

y2(1 + t3).


Other Ways of Looking at Differential Equations

1. One way that psychologists study how people learn is by asking them to learn a list of nonsensewords.(1) Suppose that L(t) is the percent of such a list that a particular person has learned at timet, where t is measured in days.

(a) A simple model for learning is to assume that learning happens at a rate proportional to thepercent of the list left to be learned. Write a differential equation that expresses this.

(b) A student who has learned a fifth of the list already is currently learning at a rate of 10% of thelist per day. Incorporate this into your model.

(c) When this student just started learning the list (so knew none of it), at what rate was he learning?

(d) How long does it take the student to go from knowing none of the list to knowing half of it?

2. In this problem, we’ll look at the differential equationdy

dt= 2t− y − 1.

(a) Which of the following is the slope field ofdy

dt= 2t− y − 1?

-2 -1 1 2t

-2

-1

1

2

y

-2 -1 1 2t

-2

-1

1

2

y

-2 -1 1 2t

-2

-1

1

2

y

(A) (B) (C)

(1)This is meant to be a “pure” experiment, in that the learners won’t have any prior knowledge that would help them withthe task.


(b) We’re interested in the solution ofdy

dt= 2t−y−1 satisfying y(0) = 1. How might we approximate

y(2)?

3. Let f(x) be the solution of the initial value problem y′ = 2 − 8xy, y(0) = 1. Use Euler’s method witha step size of 1

4 to approximate f(1).

4. Unfortunately, differential equations are often very difficult to solve; most methods only work for specifictypes of differential equations. Many differential equations end up being solved by a process of trial anderror. One of the more common “methods” for solving a differential equation is to make a rough guessabout the form of the solutions and then check that guess.

(a) Which of the following seems most likely to be a solution of the differential equation y = y′ +3x2?

i. Some sort of trigonometric function.

ii. A polynomial of degree .

iii. Some sort of exponential function.

(b) Find all solutions of the form that you picked in (a).


5. According to some sources, the longest straight road in the U.S. is North Dakota Highway 46, whichis 123 miles long. Imagine that a car is driving along ND 46; its acceleration at time t (measured inhours) is given by 5 + 2t miles/hr2.

(a) If s(t) gives the car’s position along ND 46 at time t, write a differential equation for s(t).

(b) How many initial conditions do you think are needed to determine s(t)? What might they repre-sent?

(c) Find the general solution of the differential equation you wrote down in (a).

(d) How many constants did your general solution have? How is this related to your answer to (b)?

6. Getting information from a differential equation. Suppose that y = f(x) is a solution to the differentialequation y′′ + y′ = −x2. Why can’t the the graph of f(x) ever be both increasing and concave up?

7. In this problem, you’ll find all solutions of the differential equation y′ = 4y−sin t by using a combinationof guesswork and substitution.

(a) It seems reasonable to think that the differential equation might have a solution involving sin tand cos t. Find all solutions of the differential equation of the form a sin t+ b cos t. (There is one.)

(b) Let f(t) be the function you found in (a). Use the substitution u = y−f(t) to solve the differentialequation y′ = 4y − sin t.


Second-Order Homogeneous Differential Equations with Constant Coefficients

1. Can you guess any solutions of x′′ − x = 0? (It may help to think of the equation as x′′ = x.) Guessas many solutions as you can, and check that your guesses are correct.

2. Can you guess any solutions of x′′ − 4x = 0? (It may help to think of the equation as x′′ = 4x.) Guessas many solutions as you can, and check that your guesses are correct.

3. In this problem, we’ll look at the differential equation x′′ + 3x′ + 2x = 0.

(a) Find all solutions of the differential equation of the form x(t) = ert. (There are two.)

(b) Let f(t) and g(t) be the solutions you found in (a). Show that C1f(t) + C2g(t) is also a solutionof x′′ + 3x′ + 2x = 0.

(c) The differential equation x′′ + 3x′ + 2x = 0 models the motion of a vibrating spring (why?).Explain, in terms of springs, why there are infinitely many solutions satisfying the initial conditionx(0) = 3.

(d) Explain, in terms of springs, why there is exactly one solution of x′′ + 3x′ + 2x = 0 satisfying theinitial conditions x(0) = 3 and x′(0) = −2. What is this solution?


(e) Based on your reasoning above, do you think C1f(t)+C2g(t) from (b) is the most general solution,or should there be even more solutions? Explain.

(f) In fact, we can use a clever substitution to solve the differential equation completely. Let f(t)

be one of the solutions you found in (a). First, use the substitution u(t) = x(t)f(t) to rewrite the

differential equation x′′ + 3x′ + 2x = 0 as a differential equation about u(t). Then, use thesubstitution v(t) = u′(t) to solve the differential equation completely.

4. Solve the differential equation y′′ + 2y′ − 15y = 0.

5. Solve x′′ + 4x′ + 4x = 0.

6. Solve x′′ = 0.


x′′ + bx′ + cx = 0, Euler’s Formula

1. Express each of the following complex numbers in the form a+bi, and show its position in the complexplane.

(a) eiπ

(b) 3eiπ/4

(c) 5eiπ/3

(d) reiθ where r > 0

2. Euler’s formula eiθ = cos θ+ i sin θ is useful for many things besides solving differential equations. Forexample, this formula makes it very easy to derive some trigonometric identities.

If A is any angle, then exponent rules tell us that

eiAeiA = e2iA.

Use this, together with Euler’s formula, to get formulas for cos 2θ and sin 2θ in terms of cos θ and sin θ.

3. In this problem, we’ll look at the differential equation x′′ + x = 0.

(a) Does this model a spring? If so, does it model a spring with or without friction?


(b) Find the solution of x′′ + x = 0 satisfying the initial conditions x(0) = 1, x′(0) = 1. (You mayexpress your answer using complex numbers.)

(c) Should x(t) be real or non-real?

(d) Use Euler’s formula to rewrite x(t).

(e) Does your solution seem consistent with your answer to (a)?

4. (a) Suppose a differential equation x′′ + bx′ + cx = 0 has general solution C1e2t + C2te

2t. Could thedifferential equation model a spring?

(b) Think of a differential equation whose general solution is C1e2t + C2te

2t.


x′′ + bx′ + cx = 0, Concluded

1. Solve x′′ + 6x′ + 13x = 0 with initial conditions x(0) = 7 and x′(0) = −1.

2. Solve x′′ − 2x′ + 5x = 0. (Give the general solution.)


3. Solve x′′ + 9x = 0.

4. (a) Solve x′′ + 8x′ + 17x = 0 with initial conditions x(0) = 1 and x′(0) = 0.

(b) Interpret (a) in terms of a spring. What is happening to the spring as time goes on?

5. In each part, decide whether the differential equation has periodic solutions. If so, what is the period?

(a) x′′ + 2x′ − 3x = 0.

(b) x′′ + 2x′ + 3x = 0.

(c) x′′ + 4 = 0.

(d) x′′ + 4x′ = 0.

(e) x′′ + 4x = 0.

(f) x′′ + 4t = 0.

(g) x′′ − 4x = 0.

Does this agree with your interpretation of the differential equations in terms of springs? (Be careful!Not all of the differential equations are of the form we’ve been discussing, and even the ones that arecannot necessarily be interpreted in terms of springs.)


Introduction to Systems of Differential Equations

1. Steve Strogatz, now a professor at Cornell, came up with the following system to model the relationshipbetween Romeo and Juliet.

Let j(t) be a measure of Juliet’s feelings for Romeo at time t, and let r(t) be a measure of Romeo’sfeelings for Juliet, with t = 0 being the time of the Capulet ball where Romeo and Juliet met. Positivevalues indicate love, and negative values indicate hate. Then, (according to Steve Strogatz, at least),the following system models the relationship between Romeo and Juliet:

dj

dt= −r

dr

dt= j

(a) According to this model, which of the following statements are true?

i. The more Romeo loves Juliet, the more strongly Juliet feels for him.

ii. The more Juliet loves Romeo, the more strongly Romeo falls for her.

iii. Juliet finds Romeo most attractive when he loves her least. When he loves her most, sheturns away.

(b) One of the following is the general solution of the system; which one?

i. j(t) = −C1e−t + C2e

t and r(t) = C1e−t + C2e

t.

ii. j(t) = −C1 sin t + C2 cos t and r(t) = C1 cos t + C2 sin t.

iii. j(t) = C1 sin t + C2 cos t and r(t) = C1 cos t + C2 sin t.

(c) Find the particular solution satisfying j(0) = 0 and r(0) = 1.


(d) For the particular solution you found in (c), graph j(t) and r(t) on the following set of axes:

t

Describe Romeo and Juliet’s relationship over time (using words, not math).

(e) Sketch the solution you found in (c) in the phase plane.

j

r

(f) Suppose that fate prevented Romeo from attending the Capulet ball. What initial condition couldyou then write for the system? Find the particular solution satisfying the initial condition. Graphit like you did in (d). Then, sketch the corresponding trajectory on your picture from (e).


(g) Others have suggested improvements to Strogatz’s model. Suppose that the following is deter-mined to be a more accurate solution trajectory. Sketch possible graphs of r(t) and j(t) on asingle set of axes (like in (d)). (Label any times you think are important on both the trajectoryand your graph.) What happens to Romeo and Juliet’s relationship over time?

-4 -3 -2 -1 1 2 3 4j

-4

-3

-2

-1

1

2

3

4r

2. A large savannah is populated primarily by lions and gazelles. Let L(t) be the population of lions attime t and G(t) be the population of gazelles at time t. Suppose that these satisfy the system

dG

dt= 0.01G

(1 − G

6000

)(G

2000− 1

)− 0.0001GL = −G [(G− 2000)(G− 6000) + 120000L]

1200000000

dL

dt= −0.1L + 0.00002GL =

(G− 5000)L

50000

(a) In the absence of gazelles, what does the model predict about the population of lions?

(b) In the absence of lions, what does the model predict about the population of gazelles?


(c) Find all equilibrium points of this system.

(d) In the phase plane, draw the equilibrium points and the solution trajectories corresponding to thesituations in (a) and (b).


Phase Plane Analysis

1. Mrs. Portwick moves from England to an uninhabited island. She plants a garden and decides thatthe two things missing in her garden are robins and worms. She decides to import some. Let w(t)represent the number of hundreds of worms on the island at time t and r(t) be the number of hundredsof robins on the island at time t. She models their interaction as follows:

dr

dt= −0.4r + 0.1wr

dw

dt= 0.3w − 0.3wr

Analyze the system.

2. On your phase plane analysis from #1, sketch trajectories with the following initial conditions:

(a) r(0) = 0, w(0) = 2.

(b) r(0) = 2, w(0) = 0.

3. Mrs. Portwick’s nephew Edwin comes to visit, sees her work, and says that he believes that the second

differential equation should bedw

dt= 0.3w − 0.03w2 − 0.3wr.

(a) What could his reasoning be?

(b) Analyze the new system.

4. On your phase plane analysis from #3, sketch trajectories with the following initial conditions:

(a) r(0) = 0, w(0) = 2.

(b) r(0) = 2, w(0) = 0.

(c) r(0) = 0, w(0) = 12.


For the following questions, assume that we are modeling Mrs. Portwick’s island using the model in #1.

5. It turns out that the solution trajectories of the system in #1 are closed loops. (This is NOT somethingwe can tell from the phase plane analysis.)

Suppose that, at some time that we’ll call t = 0, Mrs. Portwick’s garden has 200 robins (r = 2) and400 worms (w = 4).

(a) Describe in words what happens over time.

(b) On the following set of axes, draw possible graphs of r(t) and w(t).

t

6. One day, Mrs. Portwick asks her butler Godfrey to do a quick survey of the garden. Godfrey findsthat there are 150 robins and 300 worms. Mrs. Portwick feels that the garden has too many worms,so she asks Godfrey to take 50 of them to use as fishing bait. What effect does this have on the robinand worm populations in the long run?


Systems and Shapes of Trajectories

1. (a) (Problem Set 33, #1) Let x(t) be the number of hundreds of beasts of species X at time t andy(t) be the number of hundreds of beasts of species Y at time t. Do a phase plane analysis of thesystem

dx

dt= 0.1x− 0.025xy = 0.025(4x− xy)

dy

dt= 0.1y − 0.02xy − 0.01y2 = 0.01(10y − 2xy − y2)

(b) Using only your sketch from (a), which of the following questions can you answer?

i. If there are currently 400 X beasts and 200 Y beasts, will the population of Y beasts everreach 300?

ii. If there are currently 500 X beasts and 600 Y beasts, will the population of X beasts ever fallbelow 300?

iii. If x(0) = 2 and y(0) = 5, what are limt→∞

x(t) and limt→∞

y(t)?

iv. If x(0) = 1 and y(0) = 1, what are limt→∞

x(t) and limt→∞

y(t)?


2. In this problem, we’ll look at the systemdx

dt= y,

dy

dt= −4x.

(a) By writing and solving a differential equation for y in terms of x, determine the exact shape ofthe solution trajectories (in the phase plane) of this system.

(b) The following diagram shows the sketch you get by doing a phase plane analysis of this system.Using this together with your answer to (a), sketch a phase portrait of the system. (This means:Sketch several representative solution trajectories in the phase plane.)

-4 -3 -2 -1 1 2 3 4x

-4

-3

-2

-1

1

2

3

4y

3. In this problem, we’ll look at the systemdx

dt= y and

dy

dt= −3y.

(a) Find the shape of the solution trajectories.

(b) Do a qualitative phase plane analysis of this system.


(c) Using your answers to (a) and (b), draw a phase portrait of the systemdx

dt= y and

dy

dt= −3y.

(d) Let’s focus on one particular solution, the one satisfying the initial conditions x(0) = 0 andy(0) = 6. What is the exact shape of the solution trajectory with these initial conditions? (Writean equation which describes the shape.)

(e) On the following set of axes, sketch the graphs of x(t) and y(t) if x(0) = 0 and y(0) = 6.

t

(f) In fact, you know enough to solve the system dxdt = y, dy

dt = −3y. (Hint: First solve for y(t).)Find the general solution.

(g) Find the solution of the system satisfying x(0) = 0 and y(0) = 6. Does your answer match thegraphs you sketched in (e)?


4. On the worksheet “Phase Plane Analysis”, we used phase plane analysis to understand the system

dr

dt= −0.1r(4 − w)

dw

dt= 0.3w(1 − r)

where w(t) represented the number of hundreds of worms on an island at time t and r(t) representedthe number of hundreds of robins on the island at time t. The sketch we got is below. The solutiontrajectories are closed curves (although we didn’t show that).

Suppose that there are currently 100 robins and 200 worms on the island. What is the largest numberof robins there will ever be on the island?

0 1 2r0

2

4

6

8

10

w


Systems Wrap-Up

In Weekly Problem #32, you were given the following scenario. Suppose that there is a large population ofpeople, and some of the people have a fatal disease. This disease is infectious, so anybody who doesn’t havethe disease is susceptible to getting it. Let I(t) be the number of people infected at time t, and let S(t) bethe number of people who are susceptible at time t. You may ignore birth and death, except for death dueto the disease, which you should include.

A reasonable system for the situation described is:

dS

dt= −0.001IS = − 0.001IS

dI

dt= 0.001IS − 0.1I = 0.001I(S − 100)

1. Do a phase plane analysis for this system.

S

I

2. If the population starts with 50 infected people and 200 susceptible people, what will happen in thelong run? Do lim

t→∞I(t) and lim

t→∞S(t) exist? If so, what are they?

3. Suppose you instead want to model an epidemic like measles, which is not fatal but confers immunityon people who’ve had it. How might you reasonably model such an epidemic?


In the next few problems, we’ll look at the systemdS

dt= −IS + 50,

dI

dt= IS − 10I.

4. Do a qualitative phase plane analysis of this system.

S

I

5. Based on your phase plane analysis, what do you think the trajectories look like? Sketch a possibletrajectory on your diagram if S(0) = 5 and I(0) = 20.

6. Using your trajectory, sketch a possible graph of I(t) if S(0) = 5 and I(0) = 20.

7. This graph shows the number of cases of measles in 2 week periods in London from 1944 to 1966. (Seehttp://www.zoo.cam.ac.uk/zoostaff/grenfell/measles.htm for the raw data and more informa-tion about it.) Does our system give the same qualitative behavior?

0

1000

2000

3000

4000

5000

6000

7000

8000

1945 1950 1955 1960 1965


Documents

Math 1b: Calculus, Series, and Di erential Equationspeople.math.harvard.edu/~jjchen/docs/spring2011-1b-worksheets.pdf · Harvard University, Spring 2011 List of Worksheets Integration